There has recently been a good deal of controversy about Landauer's Principle, which is often stated as follows: The erasure of one bit of information in a computational device is necessarily accompanied by a generation of kT ln 2 heat. This is often generalised to the claim that any logically irreversible operation cannot be implemented in a thermodynamically reversible way. John Norton (2005) and Owen Maroney (2005) both argue that Landauer's Principle has not been shown to hold in general, and Maroney offers a method that he claims instantiates the operation Reset in a thermodynamically reversible way.In this paper we defend the qualitative form of Landauer's Principle, and clarify its quantitative consequences (assuming the second law of thermodynamics). We analyse in detail what it means for a physical system to implement a logical transformation L, and we make 1 this precise by defining the notion of an L-machine. Then we show that logical irreversibility of L implies thermodynamic irreversibility of every corresponding L-machine. We do this in two ways. First, by assuming the phenomenological validity of the Kelvin statement of the second law, and second, by using information-theoretic reasoning.We illustrate our results with the example of the logical transformation 'Reset', and thereby recover the quantitative form of Landauer's Principle.

Bipartite entanglement is one of the fundamental quantifiable resources of quantum information theory. We propose a new application of this resource to the theory of quantum measurements. According to Naimark's theorem any rank 1 generalised measurement (POVM) M may be represented as a von Neumann measurement in an extended (tensor product) space of the system plus ancilla. By considering a suitable average of the entanglements of these measurement directions and minimising over all Naimark extensions, we define a notion of entanglement cost E_{\min}(M) of M. We give a constructive means of characterising all Naimark extensions of a given POVM. We identify various classes of POVMs with zero and non-zero cost and explicitly characterise all POVMs in 2 dimensions having zero cost. We prove a constant upper bound on the entanglement cost of any POVM in any dimension. Hence the asymptotic entanglement cost (i.e. the large n limit of the cost of n applications of M, divided by n) is zero for all POVMs. The trine measurement is defined by three rank 1 elements, with directions symmetrically placed around a great circle on the Bloch sphere. We give an analytic expression for its entanglement cost. Defining a normalised cost of any $d$-dimensional POVM by E_{\min} (M)/\log_2 d, we show (using a combination of analytic and numerical techniques) that the trine measurement is more costly than any other POVM with d>2, or with d=2 and ancilla dimension 2. This strongly suggests that the trine measurement is the most costly of all POVMs.

Homotopy Type Theory (HoTT) is a putative new foundation for mathematics grounded in constructive intensional type theory, that offers an alternative to the foundations provided by ZFC set theory and category theory. This paper explains and motivates an account of how to define, justify and think about HoTT in a way that is self-contained, and argues that it can serve as an autonomous foundation for mathematics.We first consider various questions that a foundation for mathematics might be expected to answer, and find that the standard formulation of HoTT as presented in the "HoTT Book" does not answer many of them. More importantly, the way HoTT is developed in the HoTT Book suggests that it is not a candidate autonomous foundation since it explicitly depends upon other fields of mathematics, in particular homotopy theory.We give an alternative presentation of HoTT that does not depend upon sophisticated ideas from other parts of mathematics, and in particular makes no reference to homotopy theory (but is compatible with the homotopy interpretation). Our elaboration of HoTT is based on a new interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory.

Homotopy type theory (HoTT) is a new branch of mathematics that connects algebraic topology with logic and computer science, and which has been proposed as a new language and conceptual framework for mathematical practice. Much of the power of HoTT lies in the correspondence between the formal type theory and ideas from homotopy theory, in particular the interpretation of types, tokens, and equalities as (respectively) spaces, points, and paths. Fundamental to the use of identity and equality in HoTT is the powerful proof technique of path induction. In the 'HoTT Book' [1] this principle is justified through the homotopy interpretation of type theory, by treating identifications as paths and the induction step as a homotopy between paths. This is incompatible with HoTT being an autonomous foundation for mathematics, since any such foundation must be able to justify its principles without recourse to existing areas of mathematics. In this paper it is shown that path induction can be motivated from pre-mathematical considerations, and in particular without recourse to homotopy theory. This makes HoTT a candidate for being an autonomous foundation for mathematics.

We describe a universal information compression scheme that compresses any pure quantum i.i.d. source asymptotically to its von Neumann entropy, with no prior knowledge of the structure of the source. We introduce a diagonalisation procedure that enables any classical compression algorithm to be utilised in a quantum context. Our scheme is then based on the corresponding quantum translation of the classical LempelZiv algorithm. Our methods lead to a conceptually simple way of estimating the entropy of a source in terms of the measurement of an associated length parameter while maintaining high fidelity for long blocks. As a by-product we also estimate the eigenbasis of the source. Since our scheme is based on the Lempel-Ziv method, it can be applied also to target sequences that are not i.i.d.